Mathematical Modelling - Turning Reality into Equations
Mathematical modelling is the process of using mathematics to describe, analyse, and make predictions about real-world situations. It is the bridge between pure mathematics and the messy, complicated world we actually live in. Every weather forecast, every bridge design, every dose of medicine, and every economic policy relies on mathematical models.
Isaac Newton's 1687 Principia Mathematica is often considered the first great mathematical model of the physical world: a small set of equations describing gravity and motion that could predict the orbits of planets with remarkable accuracy. The SIR model of infectious disease taught below has its own specific origin – it was introduced in 1927 by the Scottish scientists William Ogilvy Kermack and Anderson Gray McKendrick in a paper titled “A Contribution to the Mathematical Theory of Epidemics”, nearly a century before the same equations were used to model the spread of COVID-19 and inform lockdown policy around the world.
What Is a Mathematical Model?
A mathematical model is a simplified mathematical representation of a real-world system. It captures the key features of the system using variables, equations, or functions, while deliberately ignoring details that are too complicated or irrelevant to the question being asked. All models are wrong to some degree – the art is making them useful.
The Mathematical Modelling Cycle
Step 1 – Identify the real-world problem. Clearly define what you are trying to understand or predict.
Step 2 – Make assumptions. Simplify by deciding which features matter and which to ignore.
Step 3 – Build the model. Translate assumptions into mathematical equations or functions.
Step 4 – Solve the model. Use mathematical techniques to find answers within the model.
Step 5 – Interpret the results. Translate the mathematical answers back into the real-world context.
Step 6 – Validate. Compare predictions with real data. If the model is inaccurate, refine the assumptions and repeat.
Example 1: Modelling Population Growth
Problem: A city has a population of 500 000 growing at 2% per year. Predict the population in 20 years.
Assumption: Growth rate stays constant; no migration; no resource limits.
Model: P(t) = P0(1 + r)t = 500 000 × (1.02)20.
Solution: P(20) = 500 000 × 1.4859 ≈ 742 974.
Interpretation: The city is predicted to grow to approximately 743 000 people in 20 years.
Limitation: In reality, growth rates change. A logistic model (which includes a carrying capacity) would be more accurate for long-term predictions.
Example 2: Modelling a Projectile
Problem: A ball is kicked at 20 m/s at 40° above the ground. Find when it lands.
Assumptions: No air resistance; flat ground; constant gravitational acceleration g = 9.8 m/s².
Model: Vertical position: y = v·sin(40°)·t − ½g·t². Set y = 0 to find landing time.
Solution: 0 = t(20·sin40° − 4.9t) → t = 0 or t = 20·sin(40°)/4.9 ≈ 2.63 s.
Interpretation: The ball lands approximately 2.63 seconds after being kicked.
Limitation: Real projectiles experience air resistance, which would reduce the flight time.
Example 3: Modelling Infectious Disease (SIR Model)
The SIR model divides a population into three groups: Susceptible, Infected, Recovered. Differential equations describe how people move between groups over time:
dS/dt = −βSI, dI/dt = βSI − γI, dR/dt = γI
Where β is the infection rate and γ is the recovery rate. The model predicted COVID-19 outbreak shapes before real data was available and was used to inform lockdown decisions worldwide.
Example 4: Modelling Supply and Demand
Economists model supply and demand with linear equations:
Demand: QD = a − bP (as price rises, demand falls).
Supply: QS = c + dP (as price rises, supply rises).
Equilibrium is where QD = QS. Solving gives the market price and quantity. This simple model underpins everything from pricing algorithms to government tax policy.
Assumptions and Limitations
Every model makes assumptions. The key skill is being explicit about them:
• What have you assumed is constant that might not be?
• What factors have you ignored?
• For what range of values is the model valid?
A model that works brilliantly for small inputs may fail completely for large ones.
Types of Mathematical Models
| Model Type | Key Feature | Example |
|---|---|---|
| Linear | Straight-line relationship | Cost = fixed cost + (variable cost × units) |
| Exponential | Constant percentage growth or decay | Population, radioactive decay, compound interest |
| Quadratic | Parabolic shape | Projectile height vs time |
| Statistical | Based on data and probability | Weather forecasting, clinical trials |
| Differential equations | Describes rates of change | SIR disease model, heat flow, fluid dynamics |
| Network / graph | Nodes connected by edges | GPS routing, social networks, supply chains |
Key Takeaways
- A model simplifies reality to make it mathematically tractable – all models involve assumptions.
- The modelling cycle: identify → assume → build → solve → interpret → validate → refine.
- Different model types suit different problems: exponential for growth, linear for proportional relationships, DEs for dynamic systems.
- Always state the limitations of your model and the conditions under which it is valid.
Practice: Building Models
You have completed the Mathematical Thinking section. Continue your maths journey with the topics below.